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Unformatted text preview: ∆ in the right or in the left
x
direction.
When the tangent of the angle at which the tangent line at point x0 to
When
the graph of function y = f ( x ) is intersecting the horizontal axis
depends on the sign of ∆ x , the derivative of f ( x ) at x0 is not defined.
In that case we can talk about the “left-hand side slope” and the
In
“right-hand side slope” or, more formally, the left-side and right-side
left-side
derivatives.
derivatives.
Notation for the right-side derivatives: f ′( x0 + 0 ) = lim
+ f ( x0 + ∆x ) − f ( x0 )
∆x f ′( x0 − 0 ) = lim− f ( x0 + ∆x ) − f ( x0 )
∆x x → x0 The left-side derivates: x → x0 12 An Example x, x ≥ 0
Consider a function y = f ( x ) = x = − x , x &lt; 0
This function has a kink at (0,0): Let us compute the right-side
Let
derivative at point x0 = 0 : f ′( 0 + 0 ) = lim+
x →0 (0,0) 0 + ∆x − 0
=1
∆x Now compute the left-side
Now
derivative at point x0 = 0 : f ′( 0 − 0 ) = lim+
x →0 0 − ∆x − 0
= −1
∆x Even if the derivative of x does not exist at x=0, the left-side
Even
and right-side derivatives do exist and are well defined.
13 Left­Sided and Right­Sided Limits
Limits are not only taken for the difference quotients in order to compute
Limits
the derivatives. We can talk in general about a limit of any function of
any variable:
any lim g ( v ) = g ( v0 ) = q0 v→ 0
v Limits are answering questions of the type, “what value does variable
q=g(v) approach as variable v approaches v0 ?
q=g(v)
In some cases the answer depends on the direction in which v
In
approaches v0
The left-side limit of q is symbolized by: g ( v0 − 0) = vlim g ( v )
The
−
→ v0
The right-side limit of q: g ( v0 + 0 ) = lim g ( v )
+
v →v0 14 Graphical Illustration of One­Sided Limits
q=g(v) 2, v &lt; 5
This is a step function: q = g ( v ) = 1, v ≥ 5
As v approaches 5 from the left, the left-side limit
As
of g(v) is 2.
of
However, as v approaches 5 from the right,
However,
the right-side limit of g(v) is 1.
the 2 The limit proper of g(v) at v=5 does not
The
exist, but the two one-sided (unequal)
limits do.
limits 1 5 v 15 Evaluation of the Limits
() To evaluate a limit of function f x at point x0 it suffices to make sure
To
that the right- and the left-side limits at this point are equal to each
other.
This would be the case e.g. for
This y = 2 + x2 However, what do we do when
However, x0 and any real x0 is not even in the domain of f ( x ) ? Consider the following function that is not defined at x=1: 1− x2
y = f ( x) =
1− x We can still evaluate the value of the limit at x=1 even if the
We
function is not defined at x=1 (since you can’t divide by zero) by
simplifying the expression:
simplifying 1 − x 2 (1 − x ) (1 + x )
y = f ( x) =
=
= 1 + x, x ≠ 1
1− x
1− x lim f ( x ) =2
x→
1 16 Ratio of Two Infinities
Consider finding the limit of f ( x ) =
Consider It is difficult to evaluate f
It
two infinities is equal to. 2x + 5
at x0 = + ∞
x +1 ( + ∞ ) since one is not sure what the ratio of Once a...
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